Question

1. a pyramid with slant height 6.8 mi whose triangular base measures 11 mi on each side. each altitude of the base measures 9.5 mi. find the surface area ______

Explanation:

Step1: Calculate base area

The base is an equilateral triangle. The area formula for a triangle is $A_{base}=\frac{1}{2}\times base\times height$. Here, base = 11 mi and height = 9.5 mi. So $A_{base}=\frac{1}{2}\times11\times9.5 = 52.25$ square - miles.

Step2: Calculate area of one lateral face

The lateral face is a triangle. The base of the lateral - face triangle is the side of the base triangle (11 mi) and the height is the slant height (6.8 mi). The area formula for a triangle is $A_{lateral}=\frac{1}{2}\times base\times height$. So $A_{lateral}=\frac{1}{2}\times11\times6.8 = 37.4$ square - miles.

Step3: Calculate total lateral area

Since a triangular pyramid has 3 lateral faces, the total lateral area $A_{lateral - total}=3\times A_{lateral}=3\times37.4 = 112.2$ square - miles.

Step4: Calculate surface area

The surface area $A$ of a pyramid is the sum of the base area and the total lateral area. So $A = A_{base}+A_{lateral - total}=52.25 + 112.2=164.45$ square - miles.

Answer:

164.45 square - miles